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Additional info for A Course in Mathematical Physics: Volume 4: Quantum Mechanics of Large Systems

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6. Let ai and bi be the ordered eigenvalues of Ia I and Ib I, and let Ii) denote the eigenvectors of a. 21), Tr ab = i,j :S; i L

20; 1» was used in the form 1<1>(abW = 1<1>(Ulal VlblW (with the polar decompositions a = Ulal and b = Vlbl). This = 1<1>(lbII/2UlaII/2IaII/2VlbII/2)12 s <1>(lbll/2UlaII/2 x lal l /2U*lbl l /2). <1>(lbll/2V*laII/2Iall/2Vlbll/2) = <1>(lbl Ulal U*)<1>(Vlbl V*lai) = <1>(lblla*I)<1>(lb*llal). Now let ab = Wlabl; then <1>(labl) = <1>(W*ab) s IlbW*11 x <1>(lal) s Ilbll<1>(lal). The first part of the inequality follows from I<1>(ab) I = I<1>(ab· 1)1 s 11111<1>(1 ab I) = <1>(1 ab I)· It is a corollary of the inequality that the norm of the mapping a -> <1>(ab) is I b II.

4 Representations with N 37 = 'XJ Yfi into itself. The elements of the center become multiples of the identity when projected onto Yf i ; they can assume different values only on different Yf i • The decomposition into factors is thus uniquely fixed by Z and consequently by n(d). The further decomposition into irreducible representations is not likewise fixed; some arbitrariness is connected with the spaces Yf ik • If, for example, Yf 1 = Yf 11 ® en = Yf 11 ® el EE> Yf 11 ® e2 EE> ... EE> Yf 11 ® en' then the choice of the basis {e;} for en remains free, since the space is the same for every choice of orthogonal basis.